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Theorem isphg 28216
Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
isphg.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isphg ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem isphg
Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ph 28212 . . 3 CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
21elin2 4028 . 2 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}))
3 rneq 5583 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
4 isphg.1 . . . . . 6 𝑋 = ran 𝐺
53, 4syl6eqr 2879 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
6 oveq 6911 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
76fveq2d 6437 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
87oveq1d 6920 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦))↑2) = ((𝑛‘(𝑥𝐺𝑦))↑2))
9 oveq 6911 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(-1𝑠𝑦)) = (𝑥𝐺(-1𝑠𝑦)))
109fveq2d 6437 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑠𝑦))))
1110oveq1d 6920 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2))
128, 11oveq12d 6923 . . . . . . 7 (𝑔 = 𝐺 → (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)))
1312eqeq1d 2827 . . . . . 6 (𝑔 = 𝐺 → ((((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
145, 13raleqbidv 3364 . . . . 5 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
155, 14raleqbidv 3364 . . . 4 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
16 oveq 6911 . . . . . . . . . 10 (𝑠 = 𝑆 → (-1𝑠𝑦) = (-1𝑆𝑦))
1716oveq2d 6921 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑥𝐺(-1𝑠𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
1817fveq2d 6437 . . . . . . . 8 (𝑠 = 𝑆 → (𝑛‘(𝑥𝐺(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑆𝑦))))
1918oveq1d 6920 . . . . . . 7 (𝑠 = 𝑆 → ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2))
2019oveq2d 6921 . . . . . 6 (𝑠 = 𝑆 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
2120eqeq1d 2827 . . . . 5 (𝑠 = 𝑆 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
22212ralbidv 3198 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))))
23 fveq1 6432 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
2423oveq1d 6920 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝑥𝐺𝑦))↑2))
25 fveq1 6432 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
2625oveq1d 6920 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2))
2724, 26oveq12d 6923 . . . . . 6 (𝑛 = 𝑁 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
28 fveq1 6432 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛𝑥) = (𝑁𝑥))
2928oveq1d 6920 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑛𝑥)↑2) = ((𝑁𝑥)↑2))
30 fveq1 6432 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛𝑦) = (𝑁𝑦))
3130oveq1d 6920 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑛𝑦)↑2) = ((𝑁𝑦)↑2))
3229, 31oveq12d 6923 . . . . . . 7 (𝑛 = 𝑁 → (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)) = (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))
3332oveq2d 6921 . . . . . 6 (𝑛 = 𝑁 → (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
3427, 33eqeq12d 2840 . . . . 5 (𝑛 = 𝑁 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
35342ralbidv 3198 . . . 4 (𝑛 = 𝑁 → (∀𝑥𝑋𝑦𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
3615, 22, 35eloprabg 7008 . . 3 ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))} ↔ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))
3736anbi2d 622 . 2 ((𝐺𝐴𝑆𝐵𝑁𝐶) → ((⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}) ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
382, 37syl5bb 275 1 ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  cop 4403  ran crn 5343  cfv 6123  (class class class)co 6905  {coprab 6906  1c1 10253   + caddc 10255   · cmul 10257  -cneg 10586  2c2 11406  cexp 13154  NrmCVeccnv 27983  CPreHilOLDccphlo 28211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-cnv 5350  df-dm 5352  df-rn 5353  df-iota 6086  df-fv 6131  df-ov 6908  df-oprab 6909  df-ph 28212
This theorem is referenced by:  cncph  28218  isph  28221  phpar  28223  hhph  28579
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